3.4 Conservation of Energy

Learning objectives

3.4A—Describe the energies present in a systemDescribe the energies present in a system.• A system composed of only a single object can only have kinetic energy.• A system that contains objects that interact via conservative forces or that can change its shape reversibly may have both kinetic and potential energies.3.4B—Describe the behavior of a system using conservation of mechanical energy principlesDescribe the behavior of a system using conservation of mechanical energy principles.• Mechanical energy is the sum of a system’s kinetic and potential energies.• Any change to a type of energy within a system must be balanced by an equivalent change of other types of energies within the system or by a transfer of energy between the system and its surroundings.• A system may be selected so that the total energy of that system is constant.• If the total energy of a system changes, that change will be equivalent to the energy transferred into or out of the system.3.4C—Describe how the selection of a system determines whether the energy of that system changesDescribe how the selection of a system determines whether the energy of that system changes.• Energy is conserved in all interactions.• If the work done on a selected system is zero and there are no nonconservative interactions within the system, the total mechanical energy of the system is constant.• If the work done on a selected system is nonzero, energy is transferred between the system and the environment. BOUNDARY STATEMENT AP Physics 1 expects students to know that mechanical energy can be dissipated as thermal energy or sound by nonconservative forces. AP Physics 1: Algebra-Based Course and Exam Description Work, Energy, and Power UNIT 3 TOPIC 3.5 Power | AP Physics 1: Algebra-Based Course and Exam Description

Start with the boundary

Before listing energy, draw or state the system boundary. Kinetic energy can belong to a moving object. Potential energy belongs to a system with an internal conservative interaction or reversible change of shape.

Classify the system

Chosen system	Energies it may contain	Why
One object only	Kinetic energy	No second object or reversible internal interaction is included to store potential energy
Interacting objects, such as object–Earth	Kinetic and gravitational potential energy	The conservative gravitational interaction is inside the system
Cart–ideal-spring system	Kinetic and elastic potential energy	Motion changes a reversible spring deformation inside the system

Assign potential energy correctly

A raised object alone does not contain gravitational potential energy; that energy belongs to the object–Earth system. Including several objects is not sufficient by itself—the system needs the relevant conservative interaction or reversible deformation to have potential energy.

Build the mechanical-energy account

A system's mechanical energy is the sum of all kinetic and potential energies included in that system.

$\begin{gathered}E_{\text{mech}}=K+U\\ \Delta E_{\text{system}}=E_{\text{transferred into system}}-E_{\text{transferred out}}\end{gathered}$

Track an internal conversion

For an object–Earth system falling without air resistance, gravitational potential energy decreases while kinetic energy increases by the same amount:

$\Delta K=-\Delta U_g,\qquad K_i+U_i=K_f+U_f.$

Energy changes form within the system, so its total mechanical energy remains constant.

Interpret conservation correctly

Conservation does not mean every energy type stays constant. One type can decrease as another increases. If the system's total energy changes, an equal amount of energy must have crossed between the system and its surroundings.

Keep total energy conserved

Energy is conserved in every interaction, but the energy assigned to a selected system can change when energy crosses its boundary. The same event can therefore have different—but consistent—energy accounts for different system choices.

Analyze one event two ways

Same falling event	What is inside?	Energy account
System = object only	The object, but not Earth	Gravity is external and does positive work; the object's kinetic energy increases
System = object + Earth	Object and conservative gravitational interaction	Gravity is internal; $U_g$ decreases while $K$ increases, with no energy transfer required across the boundary

Test mechanical-energy constancy

The selected system's total mechanical energy is constant when both conditions hold: (1) work done on the system from outside is zero, and (2) there are no nonconservative interactions within the system. If external work is nonzero, that work transfers energy between the system and environment.

Account for dissipation

With friction or air resistance, mechanical energy can be dissipated as thermal energy or sound. Mechanical energy may decrease, but energy itself is not destroyed; a complete account includes the transferred or transformed energy.

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